Cholesky Decomposition for the Vasicek Interest Rate Model Cholesky Decomposition for the Vasicek Interest Rate Model Dr. Muhannad Al-Saadony, Iraq Dr. Paul Hewson, UK Dr. Julian Stander, UK 10 - 03 - 2014
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Cholesky Decomposition for the Vasicek InterestRate Model
Dr. Muhannad Al-Saadony, IraqDr. Paul Hewson, UKDr. Julian Stander, UK
10� 03� 2014
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Outline
Vasicek Interest Rate Model
Bayesian Inference
Markov chain Monte Carlo McMC
The Cholesky Decomosition
Simulation study
Conclusion and Further work
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Outline
Vasicek Interest Rate Model
Bayesian Inference
Markov chain Monte Carlo McMC
The Cholesky Decomosition
Simulation study
Conclusion and Further work
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Outline
Vasicek Interest Rate Model
Bayesian Inference
Markov chain Monte Carlo McMC
The Cholesky Decomosition
Simulation study
Conclusion and Further work
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Outline
Vasicek Interest Rate Model
Bayesian Inference
Markov chain Monte Carlo McMC
The Cholesky Decomosition
Simulation study
Conclusion and Further work
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Outline
Vasicek Interest Rate Model
Bayesian Inference
Markov chain Monte Carlo McMC
The Cholesky Decomosition
Simulation study
Conclusion and Further work
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Outline
Vasicek Interest Rate Model
Bayesian Inference
Markov chain Monte Carlo McMC
The Cholesky Decomosition
Simulation study
Conclusion and Further work
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Vasicek Interest Rate Model
Vasicek Interest Rate Model
The Vasicek Interest Rate Model was introduced by OldrichVasicek in 1977. This model determines the evolution of interestrate in short-term by using a stochastic di↵erential equations
dr
t
= ✓1
(✓2
� r
t
)dt + ✓3
dW
t
where:
r
t
is an interest rate process at continuous time t,
✓1
is a parameter describing the speed of reversion,
✓2
is a parameter describing the long run mean interest rate,
✓3
is a parameter describing the instantaneous short-ratevolatility and
dW
t
is a standard Brownian motion increment over timeinterval dt.
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Vasicek Interest Rate Model
Vasicek Interest Rate Model
The Vasicek Interest Rate Model was introduced by OldrichVasicek in 1977. This model determines the evolution of interestrate in short-term by using a stochastic di↵erential equations
dr
t
= ✓1
(✓2
� r
t
)dt + ✓3
dW
t
where:
r
t
is an interest rate process at continuous time t,
✓1
is a parameter describing the speed of reversion,
✓2
is a parameter describing the long run mean interest rate,
✓3
is a parameter describing the instantaneous short-ratevolatility and
dW
t
is a standard Brownian motion increment over timeinterval dt.
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Vasicek Interest Rate Model
Vasicek Interest Rate Model
The Vasicek Interest Rate Model was introduced by OldrichVasicek in 1977. This model determines the evolution of interestrate in short-term by using a stochastic di↵erential equations
dr
t
= ✓1
(✓2
� r
t
)dt + ✓3
dW
t
where:
r
t
is an interest rate process at continuous time t,
✓1
is a parameter describing the speed of reversion,
✓2
is a parameter describing the long run mean interest rate,
✓3
is a parameter describing the instantaneous short-ratevolatility and
dW
t
is a standard Brownian motion increment over timeinterval dt.
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Vasicek Interest Rate Model
Vasicek Interest Rate Model
The Vasicek Interest Rate Model was introduced by OldrichVasicek in 1977. This model determines the evolution of interestrate in short-term by using a stochastic di↵erential equations
dr
t
= ✓1
(✓2
� r
t
)dt + ✓3
dW
t
where:
r
t
is an interest rate process at continuous time t,
✓1
is a parameter describing the speed of reversion,
✓2
is a parameter describing the long run mean interest rate,
✓3
is a parameter describing the instantaneous short-ratevolatility and
dW
t
is a standard Brownian motion increment over timeinterval dt.
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Vasicek Interest Rate Model
Vasicek Interest Rate Model
The Vasicek Interest Rate Model was introduced by OldrichVasicek in 1977. This model determines the evolution of interestrate in short-term by using a stochastic di↵erential equations
dr
t
= ✓1
(✓2
� r
t
)dt + ✓3
dW
t
where:
r
t
is an interest rate process at continuous time t,
✓1
is a parameter describing the speed of reversion,
✓2
is a parameter describing the long run mean interest rate,
✓3
is a parameter describing the instantaneous short-ratevolatility and
dW
t
is a standard Brownian motion increment over timeinterval dt.
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Bayesian Inference
Bayes Theorem
Bayes Theorem updates previous information about the paramterscalled prior information, to obtain current parameter informationcalled posterior information. The general formula is:
f (x2
|x1
) =f (x
1
|x2
)f (x2
)Rx
2
f (x1
|x2
)f (x2
)dx2
f (x2
|x1
) is the conditional probability of x2
given x
1
,
f (x1
|x2
) is the conditional probability of x1
given x
2
,
f (x2
) is the marignal probability of x2
.
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Bayesian Inference
Bayes Theorem
Bayes Theorem updates previous information about the paramterscalled prior information, to obtain current parameter informationcalled posterior information. The general formula is:
f (x2
|x1
) =f (x
1
|x2
)f (x2
)Rx
2
f (x1
|x2
)f (x2
)dx2
f (x2
|x1
) is the conditional probability of x2
given x
1
,
f (x1
|x2
) is the conditional probability of x1
given x
2
,
f (x2
) is the marignal probability of x2
.
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Bayesian Inference
Bayes Theorem
Bayes Theorem updates previous information about the paramterscalled prior information, to obtain current parameter informationcalled posterior information. The general formula is:
f (x2
|x1
) =f (x
1
|x2
)f (x2
)Rx
2
f (x1
|x2
)f (x2
)dx2
f (x2
|x1
) is the conditional probability of x2
given x
1
,
f (x1
|x2
) is the conditional probability of x1
given x
2
,
f (x2
) is the marignal probability of x2
.
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Bayesian Inference
Application to the Vasicek Interest Rate Model
Application to the Vasicek Interest Rate Model
The posterior density of the Vasicek Interest Rate Model is
p(✓1
, ✓2
, ✓3
|r) _ L(r |✓1
, ✓2
, ✓3
)p(✓1
, ✓2
, ✓3
)
p(✓1
, ✓2
, ✓3
|r) is the posterior density of the Vasicekparameters given interest rate data r = (r
1
, r2
, . . . , rT
),
L(r |✓1
, ✓2
, ✓3
) is the likelihood for data model, and
p(✓1
, ✓2
, ✓3
) is the prior of ✓1
, ✓2
and ✓3
.
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Bayesian Inference
Application to the Vasicek Interest Rate Model
Application to the Vasicek Interest Rate Model
The posterior density of the Vasicek Interest Rate Model is
p(✓1
, ✓2
, ✓3
|r) _ L(r |✓1
, ✓2
, ✓3
)p(✓1
, ✓2
, ✓3
)
p(✓1
, ✓2
, ✓3
|r) is the posterior density of the Vasicekparameters given interest rate data r = (r
1
, r2
, . . . , rT
),
L(r |✓1
, ✓2
, ✓3
) is the likelihood for data model, and
p(✓1
, ✓2
, ✓3
) is the prior of ✓1
, ✓2
and ✓3
.
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Bayesian Inference
Application to the Vasicek Interest Rate Model
Application to the Vasicek Interest Rate Model
The posterior density of the Vasicek Interest Rate Model is
p(✓1
, ✓2
, ✓3
|r) _ L(r |✓1
, ✓2
, ✓3
)p(✓1
, ✓2
, ✓3
)
p(✓1
, ✓2
, ✓3
|r) is the posterior density of the Vasicekparameters given interest rate data r = (r
1
, r2
, . . . , rT
),
L(r |✓1
, ✓2
, ✓3
) is the likelihood for data model, and
p(✓1
, ✓2
, ✓3
) is the prior of ✓1
, ✓2
and ✓3
.
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Markov chain Monte Carlo (McMC)
Markov chain Monte Carlo (McMC) Algorithm
The aim of the McMC algorithm is to determine the posteriordensity for the parameters of the vasicek Interest Rate Model byusing Metropoils-Hasting algorithm which is describing as follows:-
Simulate a candidate value ✓(⇤) from a proposaldensity k(✓(⇤)|✓(t�1))
Compute the ratio
R =p(✓(⇤)|r)k(✓(t�1)|✓(⇤))p(✓(t�1)|r)k(✓(⇤)|✓(t�1))
Compute the acceptance probability ↵ = min[R , 1].
Sample a value ✓(t) such that ✓(t) = ✓(⇤) with probability ↵;otherwise ✓(t) = ✓(t�1).
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Markov chain Monte Carlo (McMC)
Markov chain Monte Carlo (McMC) Algorithm
The aim of the McMC algorithm is to determine the posteriordensity for the parameters of the vasicek Interest Rate Model byusing Metropoils-Hasting algorithm which is describing as follows:-
Simulate a candidate value ✓(⇤) from a proposaldensity k(✓(⇤)|✓(t�1))
Compute the ratio
R =p(✓(⇤)|r)k(✓(t�1)|✓(⇤))p(✓(t�1)|r)k(✓(⇤)|✓(t�1))
Compute the acceptance probability ↵ = min[R , 1].
Sample a value ✓(t) such that ✓(t) = ✓(⇤) with probability ↵;otherwise ✓(t) = ✓(t�1).
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Markov chain Monte Carlo (McMC)
Markov chain Monte Carlo (McMC) Algorithm
The aim of the McMC algorithm is to determine the posteriordensity for the parameters of the vasicek Interest Rate Model byusing Metropoils-Hasting algorithm which is describing as follows:-
Simulate a candidate value ✓(⇤) from a proposaldensity k(✓(⇤)|✓(t�1))
Compute the ratio
R =p(✓(⇤)|r)k(✓(t�1)|✓(⇤))p(✓(t�1)|r)k(✓(⇤)|✓(t�1))
Compute the acceptance probability ↵ = min[R , 1].
Sample a value ✓(t) such that ✓(t) = ✓(⇤) with probability ↵;otherwise ✓(t) = ✓(t�1).
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Markov chain Monte Carlo (McMC)
Markov chain Monte Carlo (McMC) Algorithm
The aim of the McMC algorithm is to determine the posteriordensity for the parameters of the vasicek Interest Rate Model byusing Metropoils-Hasting algorithm which is describing as follows:-
Simulate a candidate value ✓(⇤) from a proposaldensity k(✓(⇤)|✓(t�1))
Compute the ratio
R =p(✓(⇤)|r)k(✓(t�1)|✓(⇤))p(✓(t�1)|r)k(✓(⇤)|✓(t�1))
Compute the acceptance probability ↵ = min[R , 1].
Sample a value ✓(t) such that ✓(t) = ✓(⇤) with probability ↵;otherwise ✓(t) = ✓(t�1).
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Cholesky Decomposition
Cholesky Decomposition Method
This method is a decomposition path of a symmetricpositive-definite matrix into the produce of a lower triangluarmatrix;
We have an estimate V from a pilot McMC run of theposterior variance-covariance matrix;
The Cholesky Decomposition gives a matrix M suchthat UT
U = V . Let M = (UT )�1. So,
MVM
�1 = MU
T
UM
T
= (UT )�1
U
T
U((UT )�1)T
= I .
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Cholesky Decomposition
Cholesky Decomposition Method
This method is a decomposition path of a symmetricpositive-definite matrix into the produce of a lower triangluarmatrix;
We have an estimate V from a pilot McMC run of theposterior variance-covariance matrix;
The Cholesky Decomposition gives a matrix M suchthat UT
U = V . Let M = (UT )�1. So,
MVM
�1 = MU
T
UM
T
= (UT )�1
U
T
U((UT )�1)T
= I .
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Cholesky Decomposition
Cholesky Decomposition Method
This method is a decomposition path of a symmetricpositive-definite matrix into the produce of a lower triangluarmatrix;
We have an estimate V from a pilot McMC run of theposterior variance-covariance matrix;
The Cholesky Decomposition gives a matrix M suchthat UT
U = V . Let M = (UT )�1. So,
MVM
�1 = MU
T
UM
T
= (UT )�1
U
T
U((UT )�1)T
= I .
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Cholesky Decomposition
Cholesky Decomposition for the Vasicek Interest RateModel
We shall sample the parameters ✓ = [✓1
, ✓2
, ✓3
] using a politMcMC run and find an estimate of the variance-covariancematrix V ;
We shall find a new parameter vector � = [�1
,�2
,�3
] by usingthe formula [�] = V
�1[✓];
We produce sample from the posterior density of � using theMetropolis-Hastings algorithm.
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Cholesky Decomposition
Cholesky Decomposition for the Vasicek Interest RateModel
We shall sample the parameters ✓ = [✓1
, ✓2
, ✓3
] using a politMcMC run and find an estimate of the variance-covariancematrix V ;
We shall find a new parameter vector � = [�1
,�2
,�3
] by usingthe formula [�] = V
�1[✓];
We produce sample from the posterior density of � using theMetropolis-Hastings algorithm.
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Cholesky Decomposition
Cholesky Decomposition for the Vasicek Interest RateModel
We shall sample the parameters ✓ = [✓1
, ✓2
, ✓3
] using a politMcMC run and find an estimate of the variance-covariancematrix V ;
We shall find a new parameter vector � = [�1
,�2
,�3
] by usingthe formula [�] = V
�1[✓];
We produce sample from the posterior density of � using theMetropolis-Hastings algorithm.
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Simluation Study
Simulation Study
We assume that
For the original model, the initial values for theparameters ✓
1
= 3, ✓2
= 1 and ✓3
= 2, and the initial value ofthe interest rate is 10.
For the transformed model, the initial values for theparameters ✓
1
= 3.834, ✓2
= 0.7533 and ✓3
= 1.3695, and theinitial value of the interest rate is 6.22.
The prior distribution of ✓1
and �1
are a gamma distribution,for ✓
2
and �2
are a normal distribution and for ✓3
and �3
arean inverse gamma distribution.
We select candidate values of ✓1
, ✓2
,�1
, and �2
are along-normal distribution and for ✓
3
and �3
are an inversegamma distribution.
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Simluation Study
Simulation Study
We assume that
For the original model, the initial values for theparameters ✓
1
= 3, ✓2
= 1 and ✓3
= 2, and the initial value ofthe interest rate is 10.
For the transformed model, the initial values for theparameters ✓
1
= 3.834, ✓2
= 0.7533 and ✓3
= 1.3695, and theinitial value of the interest rate is 6.22.
The prior distribution of ✓1
and �1
are a gamma distribution,for ✓
2
and �2
are a normal distribution and for ✓3
and �3
arean inverse gamma distribution.
We select candidate values of ✓1
, ✓2
,�1
, and �2
are along-normal distribution and for ✓
3
and �3
are an inversegamma distribution.
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Simluation Study
Simulation Study
We assume that
For the original model, the initial values for theparameters ✓
1
= 3, ✓2
= 1 and ✓3
= 2, and the initial value ofthe interest rate is 10.
For the transformed model, the initial values for theparameters ✓
1
= 3.834, ✓2
= 0.7533 and ✓3
= 1.3695, and theinitial value of the interest rate is 6.22.
The prior distribution of ✓1
and �1
are a gamma distribution,for ✓
2
and �2
are a normal distribution and for ✓3
and �3
arean inverse gamma distribution.
We select candidate values of ✓1
, ✓2
,�1
, and �2
are along-normal distribution and for ✓
3
and �3
are an inversegamma distribution.
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Simluation Study
Simulation Study
We assume that
For the original model, the initial values for theparameters ✓
1
= 3, ✓2
= 1 and ✓3
= 2, and the initial value ofthe interest rate is 10.
For the transformed model, the initial values for theparameters ✓
1
= 3.834, ✓2
= 0.7533 and ✓3
= 1.3695, and theinitial value of the interest rate is 6.22.
The prior distribution of ✓1
and �1
are a gamma distribution,for ✓
2
and �2
are a normal distribution and for ✓3
and �3
arean inverse gamma distribution.
We select candidate values of ✓1
, ✓2
,�1
, and �2
are along-normal distribution and for ✓
3
and �3
are an inversegamma distribution.
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Simluation Study
Simulation study for the Original Model
0 6000
3.0
3.5
4.0
Iteration t
θ 1(t)
0 10 20
0.0
0.4
0.8
Lag
ACF
θ1(t)
●●●
●●
●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●
●●●
●●●●●
●●●●●● ●●●●●●●●●●●
●●●●●●
●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●
●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●
●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●
●●●●●●●●●
●●●●
●●●●●●●●●● ●●●●●●●
●●●●●●●●●●●● ●●●●● ●●●●●●●
●
●●●
●●●
●●
●●●●●●●
●
●●●●
● ●●●●●●●●●●●●●●●● ●●●●●●●●●
● ●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●
●●●●● ●●●
●●●●●●●●●●●●●●●●●●●● ●●●●
●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●
●●
●●●●●●●●●●●●●
●●●●
●●●●● ●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●
●
●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●
●●●
●
●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●
●●●●● ●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●
●●●●●●●● ●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●
●
●●●●●●●●●
●●●●●●●
●●
●●●●●●●●●
●●● ●●●●●●●
●●
●●●●●●●●
●●●●●●●●●●●●●
●●●●●●
●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●
●●●●●●●●●
●●●●●
●● ●●
●●●●●●●●●●●●●●●●●
●●●
●●●●●●●
●●●●
●●●●●●●●●●●●●●●●●●
●●
●●●●●●●●●●●●●●●●
●●●●●●●●●●●
●●●
●●●●●●●●●●●● ●●
●●●●●●●●●●●●
●●●●●●●●●●●●●●●● ●●
●●●●●●●
●●●●●●●●●●
●●●●●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●
●●●●●●●●●
●●●●●●●●●●●●●●●
●●●●●●
●
●●●●●● ●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●
●●●●●●●
●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●
●●●●●●●●●
●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●● ●●●●●●●●●●●●●●●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●
●●
●●●●●●●●●●●
●●●●●●●●●●●●
●●●●●●
●
●●
●●●●●● ●●●●●● ●●
●●●●●●●●●●
●●●●
●●●●●●
●●●●●
●●●●●●●●●●●●●●●●
●
●●●●●
●●●●●●●●●●●●
●●●●
●●●●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●
●●●●●●●●●●●●●●
●●●●●●
●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●
●●
●●●●●
●●●●●●●●●●●●●●
●
●
●
●●●●●
●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●● ●●●●●●●●●●●●●
● ●●●●●●●●●●●●●●●●●●●●
●●●●
●●●●●●●●●●●●●●●● ●
●●
●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●
●●●●● ●●
●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●
●●●
●●●●●●●●●●●●
● ●●●●●●●●● ●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●
●●●●●●●●●
●●●●●●●●●●●●●●●●●
●
●●●
●●●
●
●●●●●●●●●● ●●●
●●
●●●●●●●●●●●
●●●
●
●
●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●
●●● ●
●●
●●●●●●●●
●●●
●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●
● ●●●●●
●●●●●●●●●●●●●
●●
●
●●●●
● ●●●●●●●●●
●●●●●●● ●●●●● ●
●●●●●●●●●●●●●●●● ●●●●
●●●●
●
●●●
●●●●●●●●
●●●
●●●●●●●●●●●●●●
●●●●●●●
●●●
●● ●
●●●●●●●
●●●●●●●●●●●● ●●●●●●●●●
●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●
●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●
●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●
●●●●●●●●●●●●●●●● ●●●●●●●●●●
●●●●●●●●●●●●●
●●●●●●●●●●●
●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●
●●●●●●●●●●●
●●●●●● ●●●●●●●●●●●●●●●●●●●●●●
●●●●●●● ●
●●●●●●●●●●●●
●●● ●●●●●●●●●●●● ●
●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●
●●●●●
●●●●●●●●●●
●●●●●
●●●●●●●●●●●●
●●●●●●●
●●
●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●
●●●● ● ●●●
●●●●●●●●●●●●●●●●●●●
●●●●●●
●●●●●●
●●●●●
●● ●●●●●●●●●●
●●●●●●●●●● ●●●●●●●●●●●●●●●●●●
●●
●●
●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●
●●●●
●●●●●
●●●●●●
●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●
●●●●●●●●●●●●●●●●●
●
● ●●●●●●●●●●●●●●●●●●●
●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●
●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●
●●●●●●
●●
●●●●
●●●●●●●
●●● ●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●
●●●●●●
●●●●●●●●●●●●●●●●●●●●●
●●●
●●●●●●
●●●●●●●
●●●●●●●●●
●●●●●
●●●●●●●●●●
●●●●●●
●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●
●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●● ●●● ●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●
●●●●●●●●●●
●
●●●●
●●●● ●●●●●●●
●●●●●●●●●●●●●●●●●●●●
●●●●●●
●●●●●●●●●●●●
●
●●●●●● ●●●●●●●●●●●●
●●●●
●●●●●●●● ●●●●●●●
●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●
●●●●
●●●●●●●●●●
●●●
●●●●●
●●● ●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●
●●●
●● ●●●●●●●●●●●●●●●●●●●●
●●●● ●●
●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●
●●●●
●
●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●● ●●●●●●●●●●●●●
●●●●●●
●●●●●●●●●●
●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●
●●●●●●●●●●●
●●●●●
●●●●●●●●●●●
●
●●●●●●●●●●●
●●●●●●
●
●●●● ●●●●●● ●●
●●●●●●●●●● ●●
●●●●● ●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●●●●●●●●
●●●●●
●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
● ●●●●●●●●●●●●●● ●●●●●●●●●●●●●
●●●●●●●●●●●●
●●●●● ●●●
●
●
●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
● ●●●●
●●●●●●●●●●●●●●●●●
●●●●
●●●●●●●●●●●●●●●●
●●●●
●
●
●●●●●●●●●●●●●●●
●●●●●●●
●●●●●●
●●●●●●●●
●
●●●●●●●●●●●
●
●●●●●●●●●●
●●●● ●●●●●●●●●●●
●●●●●●
●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●
●●●●●●●●●● ●●●●
●●●●●●●●
●●●●●●●●●●●●●●
●
●●●●●●●●●●
●●●●●● ●●●●●
●●●●●●●●
●●
●●●●●●●●●●●●
●●
●●
●●●●●●●●●
●●●●●●
●
●●●●●●●●
●
●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●●●●●●●●●●●●●●●●●●●
●●●
●●
●●●●●●●●●● ●●●●●●
●
●●●●●●● ●
●●●●●●●
●●●●●●● ●●●●● ●●
●●●●●●●●●●●
●●●
●●●
●●●●
● ●●●●●
●●●●●●●●●●●●●●●●●
●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●
●●
●●●●●●●●●●●●●●●
●●●●
●●
●
●●●
●●●●●●●●●●●●
●●●
●●●●●●●●●●●●●●●●●●●●
●●●●●●●●
●●
● ●●●●●●●●●
●●
●●●●●
●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●
●●●●●●●●● ●●●●
●●●●
●●●●●●●●●●●
●●●●●●●●● ●
●●●●●●●●●●●
●●●●
●●●●●●●●●●●●●●●●
●●●●●●●●
●●●●●●●●●●●●●●●●●●●● ●●●
●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●
●●●●●●
●●
●●●
●●●●●●●●●●●●●●●●●●●●● ●
●●●●●●●●●
●●●
●●●●●●●●●●●●●●●●●
●●●
●●
●●●●●
3.0 3.60.95
1.10
1.25
θ1(t)
θ 2(t)
0 60000.9
1.1
1.3
Iteration t
θ 2(t)
0 10 20
0.0
0.4
0.8
Lag
ACF
θ2(t)
●●●
●●●
●●●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●
●●●●●●●●●●●●●
●●
●●●●
●●
●●●●●●●●●●●●●●●●●●●● ●
●●●●●●
●●●●●●●●●●●●●●
●●●
●●●
●●●●●●
●●
●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●
●●●
●●●●●● ●●●●● ●
●
●
●●●●●●
●●●●
●●●●●●●●
●
●●●●
●●● ●●●●●
●●●●
●●●●●●● ●●
●●●●
●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●
●●●●●●●●● ●●●●●●●
●●●
●
●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●●●●●●●●●
●●●●●●
● ●●● ●
●●●●●●●●● ●●●
●
●●
●●
●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●
●●●
●●●●●●●●●●
●●●
●●●●●●
●●●●
●●
●
●
●●
●●●●●●●●●
●●●●●●●●●
●●
●
●●●●●●●●●●●●
●●●●●●●●●●●●●● ●●●●●●●
●●
●●●●●●●●
●●●●●●
●●●●●●●●●●●
●●● ●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●
●●●●
●●●●●●●
●●●●●●●
●●●
●●●
●●●●●●●
●●●●●●●●●●●●
●
●●●●●●●●●●●●●●●●●
●●●●
●●●
●●●●●●●●●●
●●●●
●●●
●●●
●●
●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●
●●
●●●●●●●●●●●●●●●●●
●●●●●
●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●
●●●●●●
●
●●●●●
●●●
●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●
●●●●
●●●●●●●●●●● ●●●
●●
●●●●●●●●● ●
●●●●●●●●●
●●●
●●● ●●●●●●●●●●
●●●●●●●●● ●●●●●
●●●●
●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●● ●●●●●●●
●●●●●●●●●●
●●●●●●●●●
●●●
●●●
●●●
●●●●●● ●●
●●●●●●●●●●●●●●●●●●●●
●●
●●●
●
●●
●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●
●●●●●●
●
●●
●
●●
●●●●●●●
●●●●●●●●
●
●
●
●●
●●●● ●●●●●●● ●
●●●●●●●●
●●●●●●● ●●● ●●●●
●●●●●●●●●●●
●●●●●●●●●●●●●●
●●●●●●●
●●●●
●●●●●●●●●●●●●●●
●●●●●
●
●
●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●
●
●●●●
●●●●●●●●●●●●●●●●●●● ●●
●●●
●
●●●●●● ●●
●●●●●●●●●●●●●●●
●●●●●●●●●
●●●●●●●●
●●●●●●●●●●
●●●●●●●●●
●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●
●●●●●●●
●●●●●●●●●●●●
●●●●
●
●●●●●●
●●●●
●●●●
●
●●
●●●●●●●●●●●●●●●●
●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●
●
●●●●●●
●●●●●●●●●●
●
●●●●●●●●●● ●●
●●●●●
●●●
●
●●● ●
●●●●●●●●●●
●●●●●●●●●●●●●●
●●●●●●●
●●●●●●●● ●●●●●●●
●●●●●●●●
●●●●
●●●●●●
●●●●●●●●●●●●●●●
●●●●●●
● ●●●●●
●●●●●●●●●●●●●●●●
●●●
●●●
●●●●
●●●●
●●●●●●●●
●●●●●●●●●
●
●●●●●
●
●●●●●●●●
●●●●●
●●●● ●●●
●●●●●●●
●●●●
●●●●
●●
●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●
●●●●
●●●●●
●●●●●●●●●●●
●●●●●●●●● ●●●●●●●
●
●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●● ●●●●●●●
●●
●●●●●●● ●●●●●●
●●●●●●●●●●●●
●●●●●●●●●●●●●●
●●●●●●
●●●
●●●●●●●● ●●●●●●
●●●●●
●●●●●●●●●●●●
●●●
●●●●●●
●●●●
●●●●●●
●●
●●● ●●●●●●●●●●●●
●●●●● ●●●●●●●●●●●●●●●●●●●
●● ●●●●●●●●
●
●●●●
●●●●●●●●
●●●
●●
●
● ●●●
●●●●●●●●●●●
●●●●●
● ●●●●●●
●●
●●●●●●●
●●●●●
●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●
●●●●
●●●●●●●●●●●
●●●●●● ●●●●●●●●●●●●●●●●●●●●
●●●●●● ●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●
●
●●●
●●●
●●●●●●●●●●●●●●●●
●
●
●●●●●●●
●●●●●●●●●●●●●●●●●●
●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●
●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●
●●●●●●●●●● ●●●●●●
●●●●●● ●●●●
●●●●●●●●●●●●
●●●●
●
●●●●
●●●●
●●●●●●●●●
●●●●●●●
●
●●
●●●
●●●
●●●●●●●
●●●●
●●●
●●●●●●●●●●●●●
●●
●●●
●●●
●●●●●●
●●●
●●●●●●●●●●●●●
●●●●●●
●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●
●●●
●●●●●●●
●●●●●●●●●●
●●●●●●●●●●●●●●
●●●●●
●
●●●●
●●●●●●●●●●●●●●●●●●●●●
●●
●●●●●●●●●●●
●●●
●
●●●●
●
●
●●●●● ● ●●●●●●●●●●●●
●●●●●
●●●●●●●●●●●●●●●●● ●
●●●● ●● ●●
●●●
●●●●●●●
●●●●●●●● ●
●●●●●●●●
●●●●●●●●●●●●●●
●●
●
●●●●●●●
●●●●●●●●●●●●
●●●
●●
●●●●●●
●●●●●●●
●●●●
●●●●●●●
●●●
●●●●●●●●●●●●●●●
●●
●●●●●●●
●●
●●●●
●●●●
●●●●●●●●●●●●●●●●●●
●●
●●
●●
●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●
●●●●●●●●●●●●●●●●
●●●
●
●●
●●●●● ●●●●●●●●●●●●
●
●●●●●●●●●●●●
● ●●●●●
●●●●●
●●●●●●●●●●
●●
●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●
●●●●●●●●●●●●●●●
●●●●●
●●●●●●●●●
●●●●●●●
●
●●●●●●●
●●●●●●●● ●●●●●●
●●
●●●●●●●●●●●
●
●●●●●●●●
●● ●●●
●
●●●●●●●●●●●●●●●●●●
●●●●●
●●●●●●
●●●●
●●●●●●●
●
●●●●
●●
●●
●●●●●●●●●●●●
●●●●●
●
●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●
●●●●●●●●●●
●●●●●
●●●●●●●
●●●●●●●●●●● ●●●●●●●
●●●
●●●
●●●●●●●●●●●●●●●●●
●●●●
●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●
●●●●● ●
●● ●●
●●●●●
●●●●
●●●●●●●●●●●
●●●●
●●●●●●●
●●●●
●●●●●
●
●●●●●●●●●●
●●●●●●●●●●●●
●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●
●●● ●
●●●●●●●●●●●●●●●●●●●
●●
●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●
● ●●●●●●●●●●●●●●●
● ●●●●
●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●
●●●●
●●●●
●● ●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●
●●●
●●●●●●●●●●● ●●●●●
●●●
●●●●●●●●●
●●●●●●● ●●●
●●●●●●●
●●●●●●●●●●●●●●●●●●
●●●●●●
●●
●
●●●●●
●●●●●●●●●
●
●●●●●●●●●●
●●●●●●●●●
●●●●●●●●●●
●
●
●●
●
● ●●●●●●●●●●
●●●
●●
●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●
●●
●●
●●●●●●●●●●●●●●●●●●
●●●
●●●●●●●●
●●
●●●●●●●●
●
●●●●●●
●●
●●
●●●●
●●●●●●
●
●●●
●●●●●●●●
●●●●
●●●●●●● ●●
●●●●●●●●●●●●●●●
●●
●●●●
●●●●●● ●●●●●●●
●●●●● ●●●●●
●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●
●●●●●●●
●●●●●●●●●●●●
●●●
●
●●●●●●●●●●●●●●●●●
●●
●●●●
●
●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●
●●●●●●●●●●
●●●●
●●●●
●●●●
●●●●●●●●●●●
●●●●
●●●●●
●●●●●●●●●●●●●
●●●●●●
●
●●●●●●● ●●●
●
●●●●●●
●●●●●●●●●●●●●●●●●●●● ●●●●●●●
●
●
●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●● ●●●●●●●●
●●●●●●●●●●●●●●●●
●●●●●●
●●
●●●●●
●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●
●●●●
●●
●●●●
●
●●●●●●
●●●●●●●●●● ●●
●●●●●●●●●●●●●●●●●●●
●●●●
●●●●●
●●
●
●
●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●● ●●●●●●
●●●●●●●●●●
●●●●●●●
●●●
●●●●●●●
●●●
●●●●●●● ●
●●●●●●●●●●●●
●●●●
●●●●●●●●●●●● ●
●●●●
●●●●●●●●●●●●●●●●●●
●
●●
●●●●
●●●●●
●
●●●●●●●●●●●●●●●●●●●●●●●●
●●●
●●●
●●●●●●●●●
●●●
●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●
●●
●●
●●● ●●
●
●●●●●●●●●●●
●●●●
●●●●●
●●●●●● ●●●●● ●●●●
●●●●●●●●●
●
●●●●
●
●●
●●● ●
●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●
●●●●●●
●●●●●●●●●●●●●●●●●●●●●●
●
●●●●●●●●●
●●●
●●●●●●●●●
●●●●●●
●●●●
●●
●●●●●●●●●●●●●●●●●●
●●
●●●●
●
●●
●●●●●
●●●●●●●●●●●●●●
●●
●●●●●●●●
●●●●●
●●
●●●●●●●●●●
●● ●●●●●●
●
●●●●●●●●●●●●●●●●●●●●●
●●●●●●
●●●●●●
●●●●●●
●●●●●●●●
●●●●●●●
●●
●●●●●●
●●●●●●●●●●●●●●●● ●●●●●●
●●●●●●●●●●●●●●●●●●●●●●
●●●●
●●
●
●●●●●●●●●
●●
●
●●●
●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●
●●●●●●●
●● ●●●●
●●
●●●●●●●●●●●●●●●●●●
●●●
●●●●●●●
●●●● ●●●●●●●●●●●●●●●●●●●
●●●●●●
●
●●●●●
●
●● ●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●
3.0 3.6
1.8
2.0
2.2
θ1(t)
θ 3(t)
0 6000
1.8
2.0
2.2
Iteration t
θ 3(t)
0 10 20
0.0
0.4
0.8
Lag
ACF
θ3(t)
●●●
●● ●
●●●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●● ●●●●● ●●●●●●●●●●●●●●●●● ●●●●●●●●
●
●●●●●●● ●●●●●●●●
●●●●●●●●●●●●●
●●
●●●●
●●
●●●●● ●●●●●●●●●●●●●●●●
●●●●●●
●●●●●●●●●●●●●●
●●●
●●●
●●●●●●
●●
●●●●●
●●●●●●●●●●●●●●●●
●●●●●●●●●●●
●●●●●●● ●●●
●●●
●●●●●●●●●●●●
●
●
●●●● ● ●
●●●●
● ●●●●●●●
●
●● ●●
●● ●●●●●●
●●●●
●●●●●●●●●
●●●●
●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●
●●●●●●●●●●●●●●●●
●●●
●
●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●
●
●●●●●●●●●
●● ●●●●
●●●●●
●●●●●●●●●●●●
●
●●
●●
●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●
●●●
●●●●●●●●●●
●●●
●●●●●●
●● ●●
●●
●
●
●●
●●●●●●●●●
●●●●●●●●●
●●
●
●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●● ●
●●
●●●●●●●●
●●●●●●
●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●
●●●●●●●
●●●●●●●
●●●
●●●
●●●●●●●
●●●●●●● ●●●●●
●
●●●●●●●●●●●● ●●●●●
●●●●
●●●
●●●●●●● ●●●
●●●●
●●●
●●●
●●
●●●●●●
●● ●●●●●●●●●●●●● ●●●●●●●●●
●●
●●●●●●●●●●●●●●●●●
●●●●●
●●●●●●● ●●●●●●●●
●●●●●● ●●●● ●●●●●●●●●●
●●●●●●
●
●●● ●●
●●●
●●●●●●●
●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●
●●●●●●●
●●●
●●● ●
●●●●●●●●●●●●●●
●●
●●●●●●●●● ●
●●●●●●●●●
●●●
●●●●● ●●●●●●● ●
●●●●●●●●● ●●●●●
●●●●
●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●● ●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●
● ●●●●●●●●●●●●●●
●●●●●●●●●●
●●●●●●●●●
●●●
●●●
●●●
●●●●●●●●
●●●●●●● ●●●●●●●●●●●●●
●●
●●●
●
●●
●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●
●●●●●●
●
●●
●
●●
●●●●●●●
●●●●●●●●
●
●
●
●●
●●●●●●●●●●●●
●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●
●●●●●●●
●●●●
●●●●●●●●●●●●●●●
●●●●●
●
●
●●●●●
●●●●●●●●●●●●●●●●●●● ●●●●●●
●●●●
●
●● ●●
●●●●●●●●●●●●●●●●● ●● ●●
●●●
●
●●●●●●●●
●●●●●●●●●● ●●●●●
●●●●●●●●●
● ●●●●●●●
●●●●●●●●●●
●●●●●●●●●
●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●
●●●●●●●
●●●●●●●●●●●●
●●●●
●
●●●●●●
●●●●
●●●●
●
●●
●●●● ●●●●●●●●●●●●
●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●● ●●●●●
●● ●●●●●●●●●●●●●●● ●●
●●●●●●●●●●●●
●
●●●●●●
●●●●●●●●●●
●
●●●●● ●●●●●●●
●●●●●
●●●
●
●● ●●
●●●●●●●●●●
●●●●●●●●● ●●●●●
●●●●●●●
●●●●●●●●●●●●●●●
●●●●●●●●
●●●●
●●●●● ●
●●●●●●●●●●●●●●●
●●●●●●
●●● ●●●
●●●●●●●●●●●●●●●●
●●●
●●●
●●●●
●●●●
●●●●●●●●
●● ●●●●●●●
●
●●●●●
●
●●●●●●● ●
●●●●●
●●●●●●●
●●●●●●●
●●●●
●●●●
●●
●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●
●●●●
●●●●●
●●● ●●●●●●● ●
●●●●●●●●●●●●●● ●●
●
●●●●●●●●●●●● ● ●●●●●●●●●●
●●●●●●●●●●●●●●●●●
●●
●●●● ●●●● ●●●●●
●●●●●●●●●●●●
●●●●●●●●●●●●●●
●●●●●●
●●●
●●●●●●● ●●●●●● ●
●●●●●
●●●●●●●●● ● ●●
●● ●
●●●●●●
●●●●
●●●●●●
●●
●●●●●●●●●●●●●●●
●●●●●●●●● ●●●●● ●●● ●●●●●●●
●●●● ●●●●●●
●
●●●●
●●●●●●●●
●●●
●●
●
●● ●●
●●●●●●●●●●●
●●●●●
●●●●●●●
●●
●●●●●●●
●●●●●
●●●●●●●●●●●●●●●
●●●● ●●●●●●●●●●●●
●●●●
●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●
●
●●●
●●●
● ●●●●●●●●●●●●●●●
●
●
●●●●●●●
●●●●●●●●●●●●●●●●●●
●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●
●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●● ●●●●●
●●●●●●●●●●●●●●●●
●●●●●●● ●●●
●●●●●●●●●●●●
●●●●
●
●●●●
●●●●
●●●●●●●●●
●●●●●●●
●
●●
●●●
●●●
●●●●●●●
●●●●
●●●
●●●●●●●●●●●●●
●●
●●●
●●●
● ●●●●●
●●●
●●●●●●●●●●●●●
●●●●●●
●●●●●● ●●●●●●●●●●
●●●●●●●●●●●●●
●●●
●●●●●●●
●●● ●●●●●●●
●●●●●●●● ●●●●●●
●●●●●
●
●●●●
●●●●●●●● ●●●●●●●●●●●●●
●●
●●●●●●●●●●●
●●●
●
●●●●
●
●
● ●●●●●●●●●●●●●●●●●
●●●●●
●●●●● ●●●●●● ●●●●●● ●
●●●●●●●●
●●●
●●●●● ●●
●●●●●●●●●
●●●●●●●●
●●●●●●●●●●● ●●●
●●
●
●●●●●●●
●●●●●●●●●●●●
●●●
●●
●●●●●●
●●●●●●●
●●●●
●●●●●●●
●●●
●●● ●●●●●●●●●●●●
●●
●●●●●●●
●●
●●●●
●●●●
●●●●●●●●●●●●●●●●●●
●●
●●
●●
●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●● ●●●●●●●●
●●●
●
●●
●●●● ●●●●●●●●●●●●●
●
●●●●●● ●●●●●●
●●●●●●
●●●●●
●●●●●●●●●●
●●
●●● ●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●
●●●●●●●● ●●●●●●●
●●●●●
●●●●●●●●●
●●●●●●●
●
●●●●●●●
●●●●●●●●●●●●●●
●●
●●●●●● ●●●●●
●
●●●●●●● ●
●●●●●
●
●●●●●●●●●●●●●●●●●●
●●●●●
●●●●●●
●●●●
●●●●●●●
●
●●●●
●●
●●
●●●●●●●●●●●●
●●●●●
●
●●●●●●●●●● ●●●●●●●●●●●●●●
●●●●●
●●●●●●●●●●
●●●●●
●●●●●●●
●●●●●●●●●●●●●●●●●●
●●●
●●●
●●●●●●●●●●●●●●●●●
●●●●
●●●●●●●●●●●●●●
●●●●●●●●●●● ●●●●
●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●
●●●●
●●●●●
●●●●
●●●●●●●●●●●
●●●●
●●●●●●●
●●●●
●●●●●
●
●●●●●●●●●●
●●●●●●●●●●●●
●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●
●● ●
●●●●
●●●●●●●●
●●●●●●●●●●●
●●
●●
●●●●●●●●●●●●●●●●●●●●● ●●●●●●
●●●●●●●●●●●●●●●●
● ●●●●
●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●● ●●●●●●●●●● ●
●●●●
●●● ●
●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●
●●●
●●●●●●●●● ●●●●●●●
●●●
●●●●●●●●●
●●●●●●●●● ●
●●●●●●●
●●●●●●●●●●●●●●●●●●
●●●●●●
●●
●
●●●●●
●●●●●●●
●●
●
●●●●●●●●●●
●●●●●●●●●
●●●●●● ●●●●
●
●
●●
●
●●●●●●●●●●●
●●●
●●
●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●
●●
●●
●●●●●●●●●●●●●●●●●●
●●●
●●●●●●●●
●●
●●●●●●●●
●
●●●●●●
●●
●●
●●●●
●●●●●●
●
●●●
●●● ●●●●●
●●●●
●●●●●●● ●●
●●●●●●●●●●●●●●●
●●
●●●●
●●●●●●●● ●●●●●
●●●●●●● ●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●
●●●●●●●
●●●●●●●●●●● ●
●●●
●
●●●● ●●●●●●●●●●●●●
●●
● ●●●
●
●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●
●●●●●●●●●●
●●●●
●●●●
●●●●
●●●●●●●●●●●
●●●●
●●●●●
●●●●●●●●●●●● ●
●●●●●●
●
●●●●●●●●●●
●
●●●●●●
●●●●●●●●●●●●●●● ●●●●●●●● ● ●●●
●
●
●●●●●●●●●●●●
●●●●●●●●●●●●●●●●● ●●●●●● ●●●●
●●●●●●●●●●●●● ●●●
●●●●●●
●●
●●●●●
●●●●●●●● ●● ●●●●●●●●●●●●
●●● ●●●●●●●●●●●●●●●●●●●●●
● ●●●
●●
●●●●
●
● ●●●●●
●●●●●● ●●●●●●
●●●●●●●●●●●●●●● ●●●●
●●●●
●●●●●
●●
●
●
●●●● ●●●●●●●●●●●●●●●● ●●
●●●●●●●●●●●● ●●
●●●●●●●●●●
●●●●●●●
●●●
● ●●●●●●
●●●
●●●●●●●●
●●●● ●●●●●●●●
●● ●●
●●●●●●●●●● ●●●
● ●●●
●●●●●●●●●●●● ● ●●●●●
●
●●
●●● ●
●●●●●
●
●●●●●●●●●●●●●●●●●●●●●●●●
●●●
●●●
●●●●●●●●●
●●●
●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●● ●●●●● ●●●
●●
●●
●●●●●
●
●●● ●●●●●●●●
●●●●
●●●●●
●●●●●●●●●●●●● ●●
●●●●●●●●●
●
●● ●●
●
●●
●● ●●
●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●● ●
●●●●●●●●●●●●●●
●●●●●●
●●●●●●●●●●●●●●●●●●●●●●
●
●●●●●●●●●
●●●
●●●●●●●●●
●●●●●●
●●●●
●●
●●●● ●●●●●●●●●●●●●●
● ●
●●●●
●
● ●
●●●●●
●●●●●●●●● ●●●●●
●●
●●●●●●●●
●●●●●
●●
● ●●●●●●●●●
● ●●●●●●●
●
●● ●●●●●●● ●●●●●●●●●●●●
●●●●●●
●●●●●●
●●●●●●
●●●●●●●●
●●●●●●●
●●
●●●●●●
●●●●●●● ●●●●●●●●●●●●●●●
●●●●●●●●●●●●● ●●●●●●●●●
● ●●●
●●
●
●●●●●●●●●
●●
●
●●●
●●●●●●●●●●●●●●●
●●●●●●●●
●●●●●●
●●●●●●●
●●●●● ●
●●●●●●●●●●●●●●●●●●●●
●●●
●●●●●●●
●●●●●●●●●●●●●● ●●●●●●●●●
●●●●●●
●
●●●●●
●
●●● ●●●●●●●●● ●●●●●●●●●●●●●
●●●●●●● ●●●●● ●●●●●
0.95 1.15
1.8
2.0
2.2
θ2(t)
θ 3(t)
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Simluation Study
Simulation study for the Modified Model
0 6000
911
1315
Iteration t
β 1(t)
0 10 20
0.0
0.4
0.8
Lag
ACF
β1(t)
●
●
●●● ●●●●● ●●●
●●●●●●●●
●●
●●● ●●●●●●●●●
●
●●●●●
●●● ●● ●●
●
●●
●● ●
●
●●●●●●●●●●●●●●●●●●
●● ●●●
●
● ●
● ●●●●●●●
●●
● ●●
●●● ●●● ● ●●●
●
●●●●
●●●●● ●●● ●● ●●●
●●●● ●●●
●●
●●●
●
●●●
●●●●●●● ● ●●●●●
●●●●●
●
●●●●●●
●●● ●●
●● ●
●●
●●
●●●●● ●●●● ●●● ●●●●● ●●●●
●●● ● ●●●●●●
●●●
●
●●● ●
●●●● ● ●●●●●●●●
●●
●
●●●●●●● ●
●●●
●●●●
●●
●●●●●●●
●●●
●
● ●●
●●
●●● ●
●●●●●●●● ●● ●●●●
●●●● ● ●●
●
●●●
●●●●●
●●●●●
●●●●●●●●●●●
●●● ●●● ●●●●●
●
● ●●●●
●●
●●
●●●●
●●●●●
●
●●●●●●●●
●●●●●
●●
●
● ●●●●●●●● ●●
●●
●●
●●●●●
●● ●●● ●●●●●●●●
●●●●●
●●●●
●●●
●●
●●●
●● ●●●●● ●
●●●●
●●● ●●●●●
●●●●●●
●●●●●●●●●● ●●●
●●●
●●●
●●
● ● ●●●●
● ●● ●●●●●●
●
●●●
●●●●●●●●●●●● ●●●●●● ●●●●●●●
●
●●●●
●
●●●●
●
●●
●
●●●●
●● ●●
●●●●●●
●●●
●
●
●● ●● ●●●●● ●●●●●● ●●
●●● ●●
●
●●
●
●●●●●●●
●●
●●●● ●●
●●●
● ●
●●●●● ●●●●●●
●●●● ●●
●●●
●
●●●
●●●●●●●●●●●
●●●●●●●●●●●●●
● ●●● ●●●●
●
●●●●
●●
●●●●●●●● ●●●
●●●
●
●●● ●●●●●
●
● ●●● ●●●●●●●●●●●●
● ●●
●●●●●
●●●●● ●●
●
●
●●●
●●●
●●●●●●●●
●
●●●●●●
●●
●● ●
●●●●
●●● ●●●●
●●● ●●● ●●●● ●
●
●● ●●●
●●●
●●
●● ●●●●●●●●
●●●●●●●●●●●●●
●●●●●
●
●●
●●●●●● ●
●●● ●
●●●
●●●●● ●
●● ●
●●●●●●●●●
●●
●● ●●●●●●●●
●●●●●
●●●●
●●●● ●●
●
●●● ●●●●●●●●●●
●●●● ●
●
●●● ●●●
●●●
●●
●●●●●●● ●●
●
●
●●●●●
● ●●● ●●● ●● ●●●●●
●●●●● ●●
●●●●●
●● ●
●
●●●●
●
●●
●●●●●●●
●● ● ●●●●●
●●
● ●●●
●●●●●●●●●
● ●
●●● ●● ●●
●● ●
●
●
●
●●●●●●● ●
●●●●●
●●●●●●●
●●●●●
●
●●●
●●●●●
●● ●●
●●● ●●●●●
●●●●●
●●●●
● ●●● ●
●●●●●●●●●●● ●●●●●●
●●●●●●●●●● ●●●● ●
●●●
● ●
●
●●● ●●●●●●● ●●●●●● ●●●●●●
●●●
● ● ●
●● ●●●●●
●●
●●●●● ● ● ●●●
●
● ●●● ●●●●●●
●●●●●●●● ●●●
●●●
●●
●
●● ●●●●●●●●● ●●●
●
●● ● ●●
●●●● ●●●
● ●
●●●
●●●●●●●●●● ●●●●●●●● ●●●
●● ●
●●●
●●
●●●●●● ●●●●● ●
●
●●● ●●●●
●●●●
●● ●●●●●●●
●
●
●●●●●●●●●●●●
●●●●●
●●● ●●●●●●●
●●●
●●●●●●
●●●
● ●
●●● ●●●●●
●
● ●
●●
●●●●●●●
●●●●● ●
●●●●●●●●●
● ●●●●● ●● ●●
●●●
●●
●●●●●
●●●●●●●● ●●●●● ●
●●●●●●●●●●●●●●●
●
●●●● ●●●●
●● ●●●
●●●●●●● ●●● ●●●● ●●●●●●
●●●●
●●● ●
●● ●● ●●
● ●●● ●●●●●●●● ●●●
●
●
●
●● ●●● ●●
●
●
●
●●●●●●●●●●
●●
●
●
●
●●●●
●●● ●●●●●●●●●●●●●●●●● ●●
● ●●●●
●●
●● ●●
●●
●●
●●●●●●●●●●●
●●●●●
●●
●●●●●●●●●●●●●● ●●● ●●●●
●● ●●●●●●
●●●●●
●
● ●●●●●●●●●
●●●●●● ●●●●●●●
●●●●●● ●●●●●●
●● ● ●
●●●●●●●●●
●●●●●● ●● ●
●●
●
●●
●●●
●●
● ●●● ●●
●●●●●●●●
●●● ●
●●●● ●●●● ●●●●● ●●
●●
●
●●●●●
●
●●
●
●
●●●●●● ●●
●●●● ●
●●●
●●●●● ●●●●●●●
●●●
● ●● ●●●●●
●●●
●●●●●●●●
●●● ●
●●
●●●●●● ●
● ●●●●●●● ●●●
●●●●●● ● ●● ●●●●
●●
● ●
●●●●●●●●●●
●●● ●●●●●●●●● ● ●
●●
●●●●●● ●●
●●●●
●●
●●●●●●●
●● ●●●●●
●●●●●●
●
●
●●●●●●
● ●●● ●●●
●●
●●●
● ●
●●
●●●
●● ●●
●●
●●
●
●●
●
●●
●
●
●●●●●●●● ●●●●●●●● ●●●
● ●●●●●●●
●●
●●
●
●●
●
●●●
●
●●●● ●●●
●●●●● ●●●●●●●
●●
●●
●
●●
●
● ●● ●●
●●●●●●●●● ●●●
●●●●
●● ●●●●●●
●● ●
●●●●●
●● ●●●●
● ●
●●●●●●●●
●●●●●●● ●
●
●●
●
●●●●●●●●●● ●● ●●●●● ●●
●●
●●●●●●●● ●●●
●●●●●●
●●●●
●●●
●● ●
●●●●●●●
●●●●
●●●●●●●
●
● ●●
● ●
●●●●● ●●●● ●
● ●● ●●●●●● ●●●
●
●●● ●●●●●●●
●
●●●
●
● ●●●●●●●●●
●●●●
●●●●●
●
●● ●
●●●●●●●●●
●●
●
●●●●●● ●
●● ●●●
●●●●● ●●
●
●
●●●●●●
● ●
● ●●● ●●
●
●●●
●●●●●●●●●●●●
●●●●●
●●● ●
●
● ●
●
●● ●●
●●●●●●●●●●●●●● ●
●●●●● ●●
● ●
●
●
● ●●●●● ●
●
●●●●
●●●●●
●
● ●● ●●●●●●●●●●
● ● ●
●●● ●●
● ●●●●●
●●
●●●●●●●●●
●●
●●●●●●●●●●●
●
●
● ●●● ●
●
●
●● ●●●●●●●●●●
● ●●●●
●●●
●●
●●●●
●●●●●●●
●●●●●●
●
●● ●●●●
●
●
●●●●●●
●●
●●●
●●●
●●●●●
●● ● ●
●
●●●
●●●●
●●●● ●
●●●●●●●●●●●
●
●
●●●
●● ●
●●●
●●●
●
●
●
●●● ●●●
●●●●● ● ●●● ●●●●●●●
●
●●●●●●
●●●
●
●● ●●
●●●● ●●● ●●●
●●
●●
●●●●●●●●●
●●●●
●●● ●●●
●●● ●●●●●●●●●
●●●●●●
●●
●
●●
●●●●●●●
●● ●
● ● ●●●
●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●
●
●
●●●● ●●
●●●●●●
●●●●●● ● ● ●●●●●● ●●●●●●
●●
●
●●●●●●●
●
●● ● ●
●● ●●
●●
●●●●●●●●●●●
●●●● ●●● ● ●●●
●●● ●● ●●●●●
●●●●
● ●●●●●●
●● ●●●●●●
●●●●●●● ●●●●
●
●●●●●●●●
●●
●●● ●●●●●●●●
●
● ●●
●●●●●●●●
●●●●●●
●●
●●●●●●●●●● ●●● ●●●
● ●
●●●●● ●●
●●
●
●●●● ●●● ●
●●●●●●●
●
●●●●● ●●●●● ●
●
●
●
●●●
●●
●
●●● ●
● ●●●
● ● ●●●
●●●●●●●
●●●●
● ●●
●
●●●
●● ●●●● ●●●● ●●●
●● ● ●●●●●●● ●●●●●●
●●● ●●●
●●●●
● ●
●●
●●●●●●●
●●●●
● ● ●
●●●●●● ●●
●●●●●●●●●●● ●●●
● ●●●●● ●●●●●● ●
●●●●●●
●●●●●
●●●●●
●
●●●●
●●●●●
●●●●●●●●●●
● ●●●●●
●●●●●●●
●● ●●●●●●●●●
●●●●●●●●
●●●● ●●●●●●●
●●●
●●● ● ●●●●●●
●●●●●●●●
●●●●●
●●●●●● ● ●
●
●●●●●● ●●
●●●●●
●●●●●●●●●
●●●
● ●●●●●●●● ●●●●●●●●● ●●●●
● ●●●●●●●●●
●●●●●●●●●●
●●● ●●●
●●
●●● ●●●●●●
● ●●● ●●●● ●●●●●●●●●●
●●
●●
●●●●● ●
●●● ●● ●●●●●●
●
● ●●●
●●●●●●
●●
●● ●
●●●●●●●
●●
● ●●● ●●
●●● ●●
●
●●●●
●●●●●●
●
●●●●●
●●●
● ●●●●● ●
●●
●●
●●●●
●●●●●●●●●
●●●●●●●●●● ●●●
●●●●
●●●●
●●●●●
●
●
● ●●●●●
● ●●●●●●●● ●●● ●● ●●
●●●
●●● ●●●●●
●●●●●
●●●
●
●●
● ●●●
●●●
●
●●
●●●● ●●●●●
●●●
●●● ●●●● ● ●●●● ●●●●● ●●●
●
●●●●
●●●
●●●●●● ●
●
●
●●
●
●●●● ●●●●●●●●●●●●
●
●●●●●●●●
●●●●●●●●●●●
●
●●●●●●●●
●●●
●
●●●●●●●●●●●●●
●●
●●● ●●
●●
●
●●
●
●●
●● ●
●●●●●●
●●●●●● ●●●
●●●● ●●●●●● ●●● ●●●
●●
●●
●●●●●● ●●
●●
●●●● ●●
●●●●●●●● ● ●●●
●●●●●●●●
●●
●●●●●●●●● ●
●●●● ●
●●●●●●●
●●●●●●●●●●●●●
●●●●●●
●● ●●●●●●
● ●●●●●●
●
●●●●
●●●
●●● ●
●●●●●
●● ●●
●●●●●●
●●● ●●
●● ●● ●●
●●●●
●● ● ●●●●●●●●●
●●●●
●●●●● ●●●● ●●●●●● ●●●●●●
●●●●
●
●●●
● ●
●●
●
●● ●●
●
●●●●●●● ●●●●●●
●●●●
●●
● ●●●●●●●●●●●●●●●●● ●●●
●●
●●●●●●● ●●●● ●●●●
●
●●●
●● ●
●●●●●● ●
●● ●
●●●●● ●●●●●●
● ●● ●●
●● ●●●●●● ●●
●●●●●●●
●●●●●●
●●●
●● ●●● ●●●
●●●●●
●
●●●● ●●●●●
●●●
●●●● ●●●●● ●●●●
●
●●●
●●
●●●●●●●●●●●●●
●●
●●●●● ●●●●
●● ●●●●
●●●●
●●●
●
●●●●●●
●●●
●●●●● ●●
●●
●
●●
●●● ●
●
●●●●●● ●●
●
●●●●●●● ●●●●
●●●●●
● ●●●●●●●●●
●●●●
●●●●●
●●●●●
●●●●●
●
●●● ●●
● ●●●
●●●●●●
●
●●●●●●●●
●
●●●●
●● ●● ●●●●●●●●●●●●● ●●●
●●●●● ●●●●●●
●●●
●
●●
●●● ●●●
●
● ●●●●● ●●●●●● ●●
●
● ●●● ● ●
●●●●●
●●
●●●
●●●●●● ●●●●●●
●● ●●●●●
●●●●●●●●●
●●●
●●●
●●
●●●●●●
●
●● ●
● ●●●
●●●● ●
●●●● ●●●●●●●● ●●●●
●●●● ●●●● ●●
●●●●
●●●
● ●●●●
●●●●●●●● ●●
●●●●● ●●●●●●● ●●● ●●●● ●● ●●●●●●●● ●●●●●●●● ●● ●●●●
●●
●●
●●
●
● ●●●
●●●●
●
●●●
●●●●
●●●●●●●● ●●●
●●●● ●●●●●●
●●●
●●●●
●●●●●●●●●●●●●●● ●●●
●●● ●
●
●●●●●●●●●●
●● ●●
●●●● ●●●●●●
●●●●
●●●●●
●● ● ●●
●●●●● ●
●
●●●●●
●
●●●
●●●●●● ●●●
●●●●● ●● ●●●●● ●●●
●● ●●
●●●●●● ●
●●●
●●
●●●●● ●●●●●●● ●●●
●
●
●●●●● ●
●●●●●
● ●●●●
● ●●●
● ●●● ●●●●●●●
●
●
●●●● ●●
●
●
●●●● ●● ●
●
●●
●●●● ●●●●●●
●●●●●● ●●●●●●●●●●●●●●
●● ●●● ●
●
●
●●●●●●●●●
●●●●●●● ●●●●●●●
●●●●●
● ●●
●●●●●
●●●●●●
●●● ●
●
●
●
●●●●
●●●●
●
●●●●●
●●●●
●● ● ●●●●●● ● ●●●●●●● ●●●●
● ●●●●●
●●●●
●●● ●●●
●
●●●●●●
●
●●●●● ●●
●●
●●●●●
●●● ●●●●●●●
●●●●●●●●● ● ● ●●●● ●●●
●
●
●● ●●●●●●● ●
●●●●●●●●● ●
●
●●●●
●●●
●
●●●
●●●●
●●●● ●●● ●●●●●●
●
●●
●●●
●
●●●●●●●●
● ●●●● ●●●●
●●●●●●●●
●●●
● ●●●●●●●●●●●●●●●●●●●●●●
●●●
●●●●●●●●●●●
●●●●●●
●● ●●●●
●●
●●●● ●●●● ●●●●●●● ● ●
●●●●●●●●●●●
●
●● ●●●
●●●
●●●●●●●●
●
●●●●●
●●●●●● ●●●●●●●●●●●●●● ●●
●● ●
●●●●●● ●●●●● ●●●●
●● ●
●
●●●
●●●
●
●●●●●●●
●
●
●●●●
●●● ●●●●●●●●●
●●● ●
●●●
●●● ●●●●●●● ●●●●●●●●
●
●
●●
●●● ●
●●
●●●
●●
●●●
●● ● ●●●●
●●●●●●●●●
●● ●●●●● ●●
●● ●●●●
●●
●●●● ●●● ●●●●●●●●●●● ●● ●●
●
●●
●●●● ●● ●
● ●●●●
●●
9 12 15
02
46
β1(t)
β 2(t)
0 6000
02
46
Iteration t
β 2(t)
0 10 20
0.0
0.4
0.8
Lag
ACF
β2(t)
●●●
●● ●●●●●
●●●●
●
●●●●●● ●●●
●
● ●●
●●●●●●●●●●●●●●●● ●● ●
●● ●●●● ●
●●●
●
●●●●●●●●
●
●●●
●●●●●
●
●
●
●● ●
● ●●●●
●
●● ●●● ●●●
●● ●●●
●
●●●●●●●
●
●●●
●● ●●● ●● ●●● ●
●●● ●●
●●
●●●●●●●●●●●●
●●● ● ●●●●
●●●●●●●
●●●●●●●●● ●
●●
● ●
●●
●●●●●●● ●●●● ●●● ●●●
●● ●●
●●
●●● ● ●●●●●●●●●●●●●●●●●● ● ●●
●●
●●●●●●●
●●●●●●●●●●●●●●●
●●●●●●●●
●●●● ●
● ●●●●●●● ●●●●
●●●
●● ●
●
●
●●●●●●● ● ●●●●
●●
●
●●●●
●●
●●●●●
●
●
●
●●●
●●●
●●● ●●●
●
●●●●
●● ●●●●
●●
●●●●●●●
●●●●
●●●●●●●●●●●●●●●●●●
●●●
●●●●● ●
●●
● ●●●●●●●
●● ●
●
● ●●●●●●●
●
●●●●●●●●●●●● ●
●●●●●● ●
●●●● ●●●●●●●● ●●●●● ●●●●●
●●●●●●
●●●●● ●●●●●●●
●
●
●●● ● ●●●●●
●●
●●●●●
●
●
●●●●●●
●
●●●●●●●●
●●●●●● ●●
●●●●●●●●●●●
●●
●●●●●●●●●●●● ●●●
●●
●●●●●●●
●●● ●● ●●●●● ●
●
●●●● ●●●●● ●●●●
●
●●
●●
●●●●●●●
●
●● ●●
●●
●●
●●●●●● ●●●●●●
●●
●● ●●●
●●●
●●●●●
●●●
●●●●●●●●● ●●●●●●●●
●
●
● ●●
● ●●
●●
●●●
●●●
●●
●●●
●●●● ●●●●●●
●
●
●● ●●●●
●●
● ●●
● ●
●●●●●
●●
●●
●●● ●●●●●●●●●
●●● ●●● ●
●●●
●●●●●●●●
●●●●●
●●●●●●
●●●
●●●
●●
●●●
●●●
●●●● ●●● ●●●● ●●●
●
●●
●●
●●●●●● ●●●
●●
●●● ●●●●
●
●●●●
●●●●●●
●
●● ●●
●
●●●●●●
●●●● ●●
●●●●●●●
● ●
● ●
●●●●●●
●●●●●●●
●●●
●●
●
●●●●●●●●
●
●
●●●●● ●●
●
●●● ●●●●●●●●●●●●●●
●●●●● ●●●●
●●●●●●●●●●●
●● ●● ●●
●
●
●
● ●●●
●●● ●● ●
●●●●●●
●●● ●●●●●●
●●●●
●●
●●●●●●●●●●●●● ●● ● ●●●●●●
●● ●
●● ●●●●
●●●●●
● ●●●● ●
● ●●●●
●●
●
●●●●●
●●●
●
●●●●●●●●●●●●●●●●●
●
●●
●●●●●
●●● ●●●●● ●●●●● ●●●●
●●●●●● ●●● ●●
●●●●●●●●●● ●●
●
●
●●●
●●●●●●●●● ●●●●
●●●
●● ●●●●● ●
●●●
●●● ●●●●●
● ●●
●●
●● ●●●● ● ●
●● ●
●●●●●●●
●●●● ●● ●●●●● ●
●●
●●●●●
●●●●●●●
●●
●●●●●●●●●●● ●●●●●●
●●● ●
●
●● ●● ● ●
●●●●● ●●●● ●●●●●●●●
●●●●●
● ●●●●
●●
●● ●●●
●● ●
●●●
●●●●●●●●
●●●●● ●●●●● ●●●●
●●●
●●● ●
●
●●●
●●●● ●●●●●●
●●●●●
● ●●●●●
●●● ●●●●
●●●●
●●●
●●
●●●●●●● ●
●●●●
●●
●
●●● ●●●
●●●●●●●●●
●●●
●
●
●●
●●●
●●●● ●●●
●
●
●
●
●
●●●●●●
●●●●
●●●●●●●●● ●●●●● ●●●
●
●●●●●●● ●
●●●
●●●●●
● ●●●●●●●●
●●●
●
●●●● ●●● ●●●● ●●●●●● ●●●●
●●● ●●●
●● ●●● ●●● ●●●
●●●●● ●●●
●●
●●●
●●● ●●● ●●●●●
●●
●●●●●●
●●●●●●
●●●
●● ●●●●●●●●●●●
●●●●
●●●●
●
●●●●●●●● ●●●●●●●●
●●●●●●●
●●●●●●●
●● ●●●●
●●●
●
●●
●
●●● ●●● ●●●●
●● ●●●●●●●
●●●●●● ●●●
●●●●●●●●●
●
●●
●●
●●●●
●
●
●●
●●● ●
●●
●
●●●
●
● ●●●●
●●●●●●
●
●
●●
●● ●
● ●●●●
●●
●●
●●●● ●●●●●●●●● ●●
●●●●● ●●●●● ●
●●● ●
●
●●● ●
●●●
●●●●
●●●●●●
●●●
●●●● ●●●●●●
●●
●●●
●●
●● ●●●
●●●
●●●
●● ●●
●●●●
●
●
●
●
●●●●●●●●●
●● ●●
●●●●●●● ●● ●●●●●●● ●●●●●●●●
● ●
●
● ●●●●●
●
● ●●●●●●●●●●
● ●●
● ●●
●●●●●●● ●
●●●●
●●●
●
● ●●●●●● ●●●
●●●
● ●●●
● ●●●●●●●●
●●●●●●●●●●●● ●
●● ●●●●● ●●●● ●●●●●
●●● ●●●●●●
●●●●●●
●● ●●●
●●●●● ●●●●
●●
●● ●●
●●●●●●●●●●●●●●
●●●●●
●●●
●●
●
●
●●
●●●●
● ●●●
●●●●●●●●● ●
●
●● ●● ●●●●●●●●●●●
●●● ●●●●●● ●●●●●●●● ●●●●●●●
● ●
●●●● ●●●●●●●●● ●●●
●●●●
●●●●● ●
●●●●●
●●●● ●
● ●
●●●● ●
●
●
● ●
●●●●●●● ●●
● ●●●●●●● ●●●●●●●
● ●●●●●●
●● ●●●●●●●●●●
●●● ●●● ●●●●●● ●●●● ●
● ●● ●
●●●●● ●●
● ●●●● ● ●●●●●●
●●● ●●● ●●●●●
●●●●
●●
●
● ●●●
●●●●● ●
●●●
●●●●
●●●●
● ●●●●●●
● ●● ●●
●
●●●●● ●●●●●●●●●
●● ●● ●●● ●●
●
●●●●●●●●●●●
●●
●●●●●●
●●
●● ●●● ●●
●● ●●●●● ●
●
●●●
●●●●●● ●●●●●
● ●●● ●●●● ●●●●● ●
●
●●●
●●
●●●●
●●
●● ●
●●●●●
●●●●● ● ●●●● ●●●
●●●●
●●●●●
●●●●●●● ●●●●
●
●●●●●
●●●●●
●
●●●●●●●●
●●●●●●
●●●●● ●●●●●●
●
●●●●●●
●
●●
●●●●
●●
●●●●●
●● ●
●
●●●●
●●
●●●●●
●●●●
●●●
●●●●
●●● ● ●●●●
●●●●●●●●● ● ●●●●●●●●●
●●●●●
●●●●
●●●●●●●●●●●●● ●●● ●
●
●●● ● ●
●● ●●●●●
●●
●●●●●●●●●●●●● ●●●
●●
●●●● ●●
●
●
●●
●
●●●●●●●
●● ●●●●●●● ●●
●
●●● ●●●●●●●
●●●●●●●● ●●●●
●●
●●
●
●●●●● ●● ● ●
●●
●
●
●
●●
●● ●●●
●●●●●●●●
●
●
●
●●
●●●
●●●●●
●
●●●
●●●●●●●●●
●●●●● ● ● ●●●●●
●
●●●●●●●●●●
●●●●●●●●●
● ●
●● ●●●● ●
●●●●●●●●●●
●
●●● ●●● ● ●
●
●
●●●
●● ●●●●●●●●●● ●●
●●●●●● ●●
●●●●●●●
●
●●● ●●●●●●●●●●●●●●●●●● ●●●●
●●●●●●
●
●●●●●●●●●●●
●●●●
●●●
●●●●
●
●●●● ●●
● ●●●● ●●●●●●
●●●
●●
●
●●●
●
●●
●●
●
●
●
●●●●
●●●●
● ●
●●
●● ●
●●●●●●●●
●●
●● ●● ●●
●● ● ●●●
●●●●
●
●● ●
●●
●●
●●● ●●●●● ●
●●● ●●●● ●●
●●● ● ●●
●●●●● ●●
●●●●
●
●
●●●●●●
●●● ●●●●●
●●●
●●●●●●● ●
●
●●●●●● ●
●●●●●●●
●●●●● ●●●●
●●●●● ●●●
●●● ●●●●●●●●●●●●●
●●●●
●●
●●●●●●●
●●●●●●●
●
●●●●
●●
●●
●●
●●●●
●
●
●● ●●●●●●●●● ●●●●●●
●●●●
●●
●●
●●●●● ●
● ●●●● ● ●●
●●●●●●●●●●●●●
●●●
● ●
●●●●●
● ●● ●●
●●●
● ●●●●●●●●●
●
●
●
●●●●●●● ●
●● ●●●●●● ●●●●●●
●●● ●●●●● ●●●●●●●●
●
●●●●●●●●●●●
●● ●●●●●●●● ●●●●●●● ●●● ●
●●
● ●
●●
●
●
●●●●●●●
●●
●
●●●● ●
●●● ●● ●●
●●●●●● ●●●●●●●●●●●●● ●●●
●●
●●●● ●● ●●● ●●●●
● ●●●●●
●● ●●●●●●● ●●●●●●●●● ●●●●● ●●●●● ●●●● ●●●●●●●●●
●
●
●●●●
●●●● ●
●●●
●●● ●●●●●●●●● ●●
● ●●●●●● ●●●
●●●●● ●●● ●
● ●●●●
●●●● ●●●●●●●●●● ●●●●●●● ●
●●
●●
●
●●●●●●● ●●
●●●●●
●●●● ●●●
● ● ●●●● ●●●●● ●●●●
●●●
●●●●●●●●●● ●●●
●●●●●●
● ●
●
●●●●●
●●●
●●● ●
●●●●●
●●
●●
●●●●●●●●
● ●●●●●●●●●●
●●●
●●●●●●●●●●●●●●●
●●● ●●●●
●
●
●●●●●● ●
●
●●●●●
●●●●●● ●●●●●●● ●●●●●
● ●●●
●●● ●
●●
●●●
●
●
●
●●●
●●●
●●● ●●●
●
●●●●●● ● ●●●●●
●●●●●●●● ●●●●●●●●● ●●●●
● ●●●●●●
●●
●
●
●
●●●
●●
●●●●●●●●●●
●
●●
●●●●
●
●● ●●
●●●● ●●●●
●
●●● ●●● ●
●●●●●●● ●●●●●●●●●
●●
●●●●
●● ●●●●●●●● ● ●●
●●
●●●●●●●●
●
●●●
●● ●●●● ●●
●●
●● ●●●
●
●● ●
●●●●●●
●● ●●●●●●
●●●
●●●●●●●
●●
●
●●●●●●●●●
●●●●●●●●●●
●●●●●●●
● ●●●●●●●
●●●●● ●●●● ●●●●● ●●● ●● ●
●●●●●● ● ●●
●●
●●●● ●●●●●
●● ●● ●●●● ●●●●●
● ●●●
●
●●●●●●●●●●●●●●●● ●●● ●●●
●●●●●● ●
●
●
●
●●●●●●●●
●●●● ●
●●●●●●●
●
●●●●
●●
●●●●
●
●
●●●●●●● ●●●●
●●● ●
●
●●
●● ●●
●●●
●●● ●●●●●●●●●●●●●●●●●● ●●
●●●●● ●●●●●
●●●●●●
●●
●●●
●●●●● ●
●●
●●●●●
●● ●●●●●●
●●●●●
●●
●●●●●●●●
●
●●
●●●● ●●●● ●●●
●●●
●●●●●
●● ●
●●●●
●●●● ●●
●● ●● ●●●
●●
●
●
●●●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●● ●●●●
● ●●●●●
●●●●●● ●
●●●
●●● ● ●
●●●●●●●
●●●●
●●●●● ●
●●●●●●● ●●●●●
●●
●●
●●●●●●●
● ●●●●●●
●●●●●●●● ●● ●●●
●●●
● ●●
●●● ●●●● ●●●● ●●●●●●●● ●●
●
● ●●●
●●●●●●● ●●●●●●●
●●●●● ●●●●●●● ●
●●
●●
●● ●●
●
●●●
● ●
● ●
●●●●
●●●●●●●
●●●
● ●● ●
●●●●
●●● ●
●
●
●●●● ●
●
●●●●●
●●●●●●●●●●
●●
● ●●●●●●● ●
●
●●●
●●
●●●●●● ●●●●●●
●●●
●●●●●
● ●●●●●
● ●●
●●●●●●●●
●●●
●
●
●●
●●● ●●
●●●●●●●●●●●●
●●●
● ●●●●
●●●
●●
●●●
●●●
●●●●
●
●
●
●● ●
●
●●●●
●● ●●
●●●●● ●●●
●● ●
●
●●●●●
● ●●●●●
●
●●●●
●●●●●●●● ●●●●● ●●●●● ●
●●
●●●● ●
●●●● ●●●● ●●● ●●●●●●
●
●
●●●●● ●
●●●●●
●● ●
● ●●
●●●●●
● ●●
●●●●●●
●●
●●
●●●●●●●●●●●
●●
●
●● ●●●
●●●●●●●●●●
●●●●●●●●● ●●●
●●●● ●●
●●
●
● ●●●●●●●●●●
●●●●●
● ●●● ●●●●●●●
●●●
●●●●● ●
●●●
●● ●
●●●●●●
● ●●●
●●●● ●●●●● ●
●●●●●●●●●●
● ●
●●●●●
●●●● ●●●●●● ●●
●●●●●●● ●●● ●●●
●●●●●●●●
●
●●
●
● ● ● ●●
●● ●●●
●●
●● ●●●●●●● ●●●
●●●●●
●● ●●●●●●●●● ●●●●●●●●
●●●● ●●
● ●
●●●
●●●●●●●● ● ●●●●●● ●●●
●●
●● ●
●●
●●●●
●●●●●
●
●●
● ●●●●●
●
●●●
●●●●●
●●●●●●●●●●● ●
●
●
●●●
●●●●●●●●●●●●● ●●●●●●●●●
● ●●●● ●
●
●
●●●● ● ●●
●●●●
●●●●●●● ●● ●●●●●●●●
●●●
●●
●●●●●●●●●●●●●
●●●●●●●●●
●●
●●● ●●●●
●
●●
●●●● ●
●●●● ●●●●●● ●●●●●●
●● ●●●●
●●●●●●
●●●●●●●
●●●●●
●●●●
●
●●
●●●●
●●● ●●●●●●●
●● ●●●
●●●●
● ●
● ●
●● ●●
● ●●●●●
●●
●
●
● ● ●●●●●●●●●●
●●●●
● ●
●●●●
●●●●
●●
●
● ●●●●●● ●●● ●●●●●●●●●●
●
●
● ●
●●●● ●
●
●
● ●● ●●
●●●●
●
●
9 12 152630
34
β1(t)
β 3(t)
0 60002630
34
Iteration t
β 3(t)
0 10 20
0.0
0.4
0.8
Lag
ACF
β3(t)
●● ●
●●●●●●●
●●● ●
●
●●●●●●●●●
●
●●●
●●●●●●●●●●●●●●●●●●●
● ● ●●●●●
● ●●
●
●●●●●●●●
●
●●●
●●●●●
●
●
●
●●●
●●●●●
●
●●●●
●●●●
●●●●●
●
●●●● ●●●
●
●●●
●●●●●●●●●●●
●●●●●
● ●
●●●● ●●●● ●●●●
●●●●●●●●
●●●●●● ●
●●●●●●●●●●
● ●
●●
●●
●● ●●●●●●●●●●●●●●●
●●●●
●●
●●●●●●●●●●●●● ●●●●●●●●●●●●
●●
●●●● ●●●
●●●●●●●●●●● ●●●●
●● ●●●●●●●●●● ●
●●● ●● ●●●●●●●
●●●
●●●
●
●
●●●●●●●●●●● ●
●●
●
●●●●
●●
●●●●●
●
●
●
●●●
●●●
●●●●●●
●
●●●●
●●●●●●
●●
●●●●●●●
●●●●
●●●●●●●●●●●●●● ●●● ●
●●●
●●●●●●
● ●
●●● ●●●●●
●●●
●
●●●●●●●●
●
●●●●● ●●●● ●●●●
●●●● ●●●
●●●●● ●●●●●●●●●●●●●●●●●
● ●●●●●
●●●●●●●●●●● ●
●
●
●●●●●●●● ●
●●
●●●●●
●
●
●●●●●●●
●●●●●●●●
●●●●●●●●
●●●●●●●●●● ●
●●
●● ●●● ●●●●● ●●●● ●
●●
●●●●●●●
●●●●●●●●●●●
●
●●●●●●●●●●●● ●
●
● ●
●●
●●●● ●●●
●
●●●●
●●
●●
●●●●●●●●●●●●
●●
●●●●●
●●●
●●●●
●
●●●
●●●●●●●●●●●●●●●●●
●
●
●●●
●●●
●●
● ●●
●●●
●●
●●●
●●●●●●● ●●●
●
●
●●●●●●
● ●
●●●
●●
●●●●●●●
●●
●● ●●●●●●●● ●●
●●●●● ●●
●●●
●●●●●●●●
●●● ●●
●●●●● ●
● ●●
●●●
●●
●●●
●●●
● ●●●●●●●●●●●●●
●
●●
● ●
●●●● ●●●●●
●●
●●●●●●●
●
●●●●
●●●● ●●
●
●●● ●
●
●●●●●●
● ●●●●●
●●●●●●●
● ●
●●
●●●●●●
●●●●● ●●
●●●
●●
●
●●●●●●● ●
●
●
●●●●●●●
●
●●●●●●●●●●●●● ●●●●●●●
●●●●●●
●● ●● ●●●●●●●
●●●● ●●
●
●
●
●●●●
●●●●●●
●●●● ●●
●●●●●●●●●
● ●●●
●●
●●●● ●●●●●●●●● ●●●●●●●●●
● ●●
●●●●●●
●●●●●
●●●●●●
●●● ●●
●●
●
● ●●●●
●●●
●
●●●●● ●●●●●●●●●●●●
●
●●
●●●●●●●●●● ●●●●●●●●●●●●
●●●●●●●●●● ●
●●●●●●●●●●●●
●
●
●●●
●●●●●●●●●●●●●
●●●
● ●●●●●●●
●●●
●●●●●●●●
●●●
●●
●●●●● ●●●
●●●
●●●● ●●●
●●●●●●●●● ● ●●
●●
●●●●●
●●●●●●●
●●
●●● ●●●●● ●●●●●●●●●
●●●●
●
●● ●●●●
●●●●●●●● ●●●●●●●●●
●●●●●
●●●●●
●●
●●●●●
●●●
●●●
●● ●●●●●●
●●●●●● ●●●●●●●●
●●●
●●●●
●
●●●
●● ● ●●●●●●●●●●●●
●●●●●●
●●●●●●●
●●●●
●● ●
●●
●●●●●● ●●
●●●●
●●
●
● ●●● ●●
●●●●●●● ●●
●●●
●
●
●●
●●●
●●● ●●●●
●
●
●
●
●
●●●● ●●
●●●●
●●●●●●●●●●●●●●● ●●
●
●●●●●●●●
●●●
● ●●●●
●●●●●●●●●
● ●●
●
●●●●●●●●●●●●●●●●●●●●●
●●●● ●●
●●●●●●●●●●●
●●●●●●●●
●●
● ●●
●●●●● ●●● ●●●
●●
●●●●● ●
●● ●●●●
●● ●
●●●●●●●●●●●●●
●●●●
●●●●
●
●●●●●● ●●●● ●●●●●●
●●●●●●●
●● ●●●●●
●●●●●●
●●●
●
●●
●
●●●●●●●●●●
●●●●●●●●●
●●●●● ●●●●
●●●●●● ●●●
●
●●
●●
●●●●
●
●
●●
●●●●
●●
●
●● ●
●
●●●●●
●●●●●●
●
●
●●
●●●
●● ●● ●
●●
●●
●●● ●●●●●●●●●●●●
●● ●●●● ●●●●●
●●●●
●
●●●●
● ●●
●●●●
●● ●●●●
● ●●
●●●●●●●●●●
● ●
●●●
●●
●●●●●
●●●
● ●●
●●●●
●●●●
●
●
●
●
●●●●●●●●●
●●●●
● ●●●●●●●●●●●●●●●●●● ●●●●●
●●
●
●●●●●●
●
●●●●●●●●●●●
● ●●
●●●
●●●●●●●●
●●● ●
●●●
●
●●●●●●●●● ●
●●●
●●●●
●●●●●● ●●●
●●● ●●●●●●●● ●●
●●●●● ●● ●●●●● ●● ●●
●●●●●●
●●●●●●● ●●
● ●●●●
●●●●●●●●●
●●
●●●●
● ●●●●●●●●●● ●●●
●●● ●●
●● ●
●●
●
●
●●
●●●●
●●●●
●●●● ●● ●● ●●
●
● ●●●●●●●●●●●●●●
●●● ●●●●●●●●●●●● ●●●●●●●● ●
●●
●●●●● ●●●●●●●●●●●
●●●●
●● ●● ●●
●●●●●
●●●●●
●●
●●●●●
●
●
●●
●●●●●●●●●
● ●●●●●●●●●●●●● ●
●●●●●●●
●●●●●● ●●●●●●
● ●●●● ●●●●●●●●●●●●
●●●●
●●●●●●●
●● ●●●●●●●●●●
●●●●● ●●●●●●
●●●●
●●
●
● ●●●
●●● ●●●
●●●
●●●●
●●●●
●●●●●●●
●●●●●
●
●●●●●●● ●● ●●●●●
●●● ●●●●●●
●
●●●●●●●●●●●
●●
●● ●●●●
●●
●●●● ●● ●
●●●● ●●●●
●
●●●
●●●●●●●●●●●
●●● ●●●●●●●●●●●
●
●●●
●●
●●●●
●●
●●●
●●●●●
●●●● ●●● ●●●●●●
●●●●
● ●●●●
●●●●●●● ●●●●
●
●●●●●
●●●● ●
●
●●●● ●●●●
●●●●●●
●●●●●●●●●●●
●
●●●●●●
●
●●
●●●●
●●
●●●●●
●●●
●
●●● ●
●●
●●●● ●
●●●●
●●●
●●●●
●●●●●● ●●
●●●●●●●●●●●●●●●●●●●
●● ● ●●
●●●●
● ●●●●●● ● ●● ●●●●●●●
●
●●●●●
●●●●●●●
●●
●●●●●●● ●●●●●●●● ●
●●
●●●●●●
●
●
●●
●
●●●●●●●
●● ●●●● ●●●●●
●
●●●●●●●●●●
●●●●●●●● ●● ●●
● ●
●●
●
●●●●●● ●●●
●●
●
●
●
●●
●●●●●
●●●●●●●●
●
●
●
●●
●●●
●●● ● ●
●
●●●
●●●●●●●●●
●●●●●●●●●●●●
●
●●●●●●●● ● ●
●●●●●●● ●●
●●
●●●●●● ●
●●●●●●●●●●
●
●●●●●●●●
●
●
●●●
●●●●●●● ●●●●●●●
●●●● ●●●●
●●●●●●●
●
●●●●●●●● ●●●●●●●●●● ●●●●●●●
●●●●● ●
●
●●●●●●●●●●●
●●●●
●● ●
●●●●
●
●●●●●●
●●●●●● ●●●●●
●●●
● ●
●
●●●
●
●●
● ●
●
●
●
●●● ●
●●●●
●●
●●
●●●
●● ● ●●● ●●
● ●
●●●●●●
● ●●●●●
●●●●
●
●●●
●●
● ●
●● ●●●● ●●●
●●●●●●●●●
●●●●●●
●●●●●●●
●●●●
●
●
●●●●●●
●● ●● ●●●●
●●●
●● ●●●●●●
●
●●●●●●●
●●●●●●●
●●●●●●●● ●
●●●●●●●●
●●●● ●●●●●●●●●●●●
●●●●
● ●
●●●●●●●
● ●●●●●●
●
●●● ●
●●
●●
●●
●●●●
●
●
●●●●●●●●●●● ●●●●●●
●● ●●
●●
●●
●●●●● ●
●●●●●●●●
●●●●●●●●●●●●●
●●●
● ●
●●●●●
●●● ●●
●●●
●●● ●●●●●●●
●
●
●
●●●● ●●●●
●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●
●
●●●●●●●●●●●
●●●●● ●●●●●●●●●●● ●●●●●
●●
●●
●●
●
●
●●●●● ●●
●●
●
●●●●●
●●●●●●●
●●●● ●●●●● ●●●●●● ●●●●●●●
●●
●●● ●●●●●●●● ●●
●●● ●●●
●●●●●●●● ●●●●●● ●●●●●●●●●● ●●●● ●●●●●●●●●●●●●
●
●
●●●●
●●●●●
●● ●
●●●●●●●●●
●●● ● ●
●●●●●● ●●●●
●●●●●●●●●
●●● ●●
●●●●●●●●● ●●●●● ●●●● ●● ●●
●●
●●
●
● ●●●●●●●●
●●● ●●
●●●●●●●
●●●●●●●●●●●●●●●
●●●
● ●●● ●●●●●●●● ●
●● ● ●●●
●●
●
●●●●●
●●●
●●● ●
●●●●●
●●
●●
●●●●●●●●
● ●●●●●●●●● ●
●● ●
●●●●●●●●●●●●●●●
●●●●●●●
●
●
●●●● ●●●
●
●●●●●
●●●●●●●●● ●●●●●●●●●
●●●●
●●●●
● ●
● ●●
●
●
●
●●●
●●●
●●●●●●
●
●●●●●●●●●● ●●
●●●●●● ●● ●●●●●●●●●●●●●
●● ●●●●●
●●
●
●
●
●●●
●●
●●●●● ●●●●●
●
●●
●●●●
●
● ●●●
●●●●● ●●●
●
●●●●●●●
●●●●●●●●●●●●●●● ●
●●
●● ●●
●●●●●●●●●●●●●
●●
●●●●● ●●●
●
●●●
●●●●●●●●
●●
●●●●●
●
●●●
●●● ●●●
● ●●●● ●●●
●● ●
●●●●●●●●●
●
●●●●●●●●●
●●●●●●●●●●
●●●●●●●
●●●●●● ●●
●●●●●●●●●●●●● ●●●● ●●●
●●●●●●●●●
● ●
●●●●●●●●●
●●●●●●●●●●●●●
●●● ●
●
●●●●●●●●●●●●●● ●●●●●●●●
●●●●●● ●
●
●
●
●●●●●●●●
●●●●●
●●●●●●●
●
●●●●
●●
●●●●
●
●
●●●●●●●●● ●●
●●●●
●
●●
●●●●
●● ●
●●● ●●●● ●●●●●● ●●● ●●●●●●●
●● ● ●●●●●● ●
●●●●●●
●●
● ●●
●●●●●●
●●
●●●●●
●●●●●●●●
●●●●●
●●
●●●●● ●●●
●
●●
●●●●● ●●●●● ●
●●●
●●●●●●● ●
●●●●
●●●● ●●
●●●●●●●
●●
●
●
●●●●●●●●●●●●●●●●●●●●●● ●●●● ●● ●●●●●●●
●●●●●●
●●●●●●●
● ●●
●●●●●
●●●●●●●
●●●●
●●●●●●
●●●●● ●●●●●●●
●●
●●
●●●●●●●
● ●●●●● ●
●●●●●● ●●●●●●●
●●●
●●●
●●●●●●●●●●●●●●● ●●●●●●
●
●●● ●
●●●●●●●●●●●●●●
●●●●●●●●●●●●●
●●
●●
●●●●
●
●●●
●●
●●
●●●●
●●●●●●●
●●●
●●●●
●●● ●
●●● ●
●
●
●●●●●
●
●●● ●●
●●●●● ●●●●●
●●
●●●●●●●●●
●
●●●
● ●
●●●●●●●●●●●●
●●●
●●●●●
●●●● ●●
●●●
●●●●●●●●
●● ●
●
●
●●
●●●●●
●●●● ●●●●●●●●
● ●●
●●●●●
●●●
● ●
●●●
●●●
●●●●
●
●
●
●●●
●
● ●●●
●●●●
●●●●●●●●
●●●
●
●●●●●
●●●●●●
●
●●●●
●●●●●●●●●●●● ● ●●●●●●
●●
●●● ●●
●●●●●●● ●●●●●●●●●●
●
●
● ●●●●●
●● ●●●
●●●
●●●
●● ●●●
●●●
●●●●●●
●●
●●
●●●●●●●●●●●
●●
●
●●●●●
● ●● ●●●●●●●
●●●●●●●●●●●●
●●●● ●●
●●
●
●●●●●●●● ●●●
●●●●●
●●● ●●●●●●●●
●● ●
●●●●● ●
●●●
●●●
●●●●●●
●●●●
●●●●●●●●●●
●●●● ●●●● ●●
●●
●● ●●●
●●●●● ●●●●●●●
●●●●●●●●●●●●●
●●●● ●●●●
●
●●
●
●●●●●
●●●●●
● ●
●●●●●●●●●●●●
●●●●●
●●●● ●●●●●●● ●●●● ●●●●
●●●●●●
●●
●●●
●●● ●●●●● ● ●●●●●●●●●
●●
●●●
●●
●●●●
●●●●●
●
●●
●●●●●●
●
●●●
●●●●●
●●●●●●●● ●●●●
●
●
●●●
●●●●●●●●●●● ●●●●●●●● ●●●
●●●●●●
●
●
●●●●●●●
●●●●
●●●●●●●●●●●● ●●● ●●
●●●
●●
● ●●●●●● ●●●●●●
●●●●●●●●●
●●
●●●●● ●●
●
●●
●●●●●
●●●●●●●● ●●● ●●●●●
●● ●●●●
●●●●●●
●●●● ●●●
●●●●●
●●●●
●
●●
●●●●
●●●●●●●●●●
●●●●●
●●● ●
●●
● ●
●●●●
● ●●●●●
●●
●
●
●●●●●● ●●●●●●
●●● ●
●●
●●●●
●●●●
●●
●
● ●●●●●●●●●●●●●●●●●●●
●
●
●●
●●●●●
●
●
●●●● ●
●●●●
●
●
0 2 4 62630
34
β2(t)
β 3(t)
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Conclusion and Further work
Conclusion and Further work
It clearly that the modified model is better than the originalmodel because the autocorrelation is less than for the orignialmodel.
The Cholesky Decomposition can readily lead to betterinference on the model.
It may be a good idea to use a real data to our suggested.
We believe that it would be extended this work into the othermodels such as the CIR model which has a similar parameterstructure.
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Conclusion and Further work
Conclusion and Further work
It clearly that the modified model is better than the originalmodel because the autocorrelation is less than for the orignialmodel.
The Cholesky Decomposition can readily lead to betterinference on the model.
It may be a good idea to use a real data to our suggested.
We believe that it would be extended this work into the othermodels such as the CIR model which has a similar parameterstructure.
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Conclusion and Further work
Conclusion and Further work
It clearly that the modified model is better than the originalmodel because the autocorrelation is less than for the orignialmodel.
The Cholesky Decomposition can readily lead to betterinference on the model.
It may be a good idea to use a real data to our suggested.
We believe that it would be extended this work into the othermodels such as the CIR model which has a similar parameterstructure.
30/08/2013 00:14
Page 1 of 1https://www.google.iq/blank.html
Cholesky Decomposition for the Vasicek Interest Rate Model
Conclusion and Further work
Conclusion and Further work
It clearly that the modified model is better than the originalmodel because the autocorrelation is less than for the orignialmodel.
The Cholesky Decomposition can readily lead to betterinference on the model.
It may be a good idea to use a real data to our suggested.
We believe that it would be extended this work into the othermodels such as the CIR model which has a similar parameterstructure.
Related Documents
